Is there a name for this construction?

[DeadWolfe]

So I was talking with a friend about a problem and noticed the following construction arose naturally:

For a category C, let F(C) be the category where objects are objects in C, arrows are finite lists of arrows, identity is the list {id}, and composition is defined by tensoring as follows:

{f_1,...f_n} comp {g_1, ... g_m} = {f_1g_1, ...f_1g_m, f_2g_1, ... f_2g_m, ... f_ng_m}

is there a name for this construction?

 

[Hurkyl]

I haven't seen it before (but that doesn't say much). In what context did it appear?